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EE301 Parallel Circuits and Kirchhoff’s Current Law

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Learning Objectives Restate the definition of a node and demonstrate how to measure voltage and current in parallel circuits Solve for total circuit resistance of a parallel circuit State and apply KCL in the analysis of simple parallel circuits Demonstrate how to calculate the total parallel resistance given various resistors connected in parallel Evaluate why homes, businesses and ships are commonly wired in parallel rather than series Demonstrate how to calculate the total current and branch currents in a parallel circuit using the current divider equation Determine the net effect of parallel combining voltage sources Compute the power dissipated by each element in a parallel circuit, and calculate the total circuit power

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Parallel Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. 2- , 3- , and 2A are in parallel

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Parallel Circuits Remember: nodes are connection points between components Notice each component has two terminals and each is connected to one of the nodes above

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Parallel Circuits House circuits contain parallel circuits The parallel circuit will continue to operate even though one component may fail open

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Parallel Circuits vs. Series Circuits In a series circuit, failure of a single components can disable all components in the circuit. In a parallel circuit, failure of one component will still allow other components to operate.

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Circuit Breaker panel in house Homes and ships are usually wired in parallel instead of series. All components can operate at rated voltage independent of other loads when wired in parallel.

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Series - Parallel Circuits Circuits may contain a combination of series and parallel components

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Parallel Circuits To analyze a particular circuit First identify the node Next, label the nodes with a letter or number Then, identify types of connections

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Example Problem 1 Determine which elements are connected in parallel and which are connected in series

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Parallel vs. Series current flow

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Kirchhoff’s Current Law (KCL) The algebraic sum of the currents entering and leaving a node is equal to zero Currents entering a node are positive and those leaving a node are negative.

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Kirchhoff’s Current Law (KCL) KCL can also be stated as “the sum of currents entering a node is equal to the sum of currents leaving the node.”

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KIRCHHOFF’S CURRENT LAW FIG. 6.31 (a) Demonstrating Kirchhoff ’s current law; (b) the water analogy for the junction in (a).

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KIRCHHOFF’S CURRENT LAW In technology, the term node is commonly used to refer to a junction of two or more branches. FIG. 6.32 Two-node configuration for Example 6.16.

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KIRCHHOFF’S CURRENT LAW FIG. 6.33 Four-node configuration for Example 6.17.

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KIRCHHOFF’S CURRENT LAW FIG. 6.34 Network for Example 6.18.

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KIRCHHOFF’S CURRENT LAW FIG. 6.35 Parallel network for Example 6.19.

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Direction of Current Assume a current direction and draw current arrows. If this assumption is incorrect, calculations will show that the current has a negative sign Negative sign simply indicates that the current flows in the opposite direction to the arrow you drew

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Example Problem 2 Determine the magnitude and direction of each current:

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Example 3 Determine the currents I 2 and I 3

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Resistors in Parallel For a circuit with 3 resistors:

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Resistors in Parallel Since voltage across all parallel elements in a circuit are the same (E = V 1 = V 2 =V 3 ):

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Resistors in Parallel For a circuit with 3 resistors:

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Resistors in Parallel Total resistance of any number of resistors in parallel:

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Current Divider Rule

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Example Problem 4 Use the current divider rule to determine all unknown currents:

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Analysis of Parallel Circuits Voltage across all branches is the same as the source voltage Determine current through each branch using Ohm’s Law Find the total current using Kirchhoff’s Current Law

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Example Problem 5 a. Determine all unknown currents and total resistance. b. Verify KCL for node a

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Power Calculations To calculate the power dissipated by each resistor, use either VI, I 2 R, or V 2 /R Total power consumed is the sum of the individual powers Compare with I T 2 R T

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Example Problem 6 a. Solve for indicated currents. b. Determine power dissipated by each resistor c. Verify total power = sum of all power dissipated

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Battery cells in series vs parallel Cells connected in series increases available voltage. Cells connected in parallel increases available current.

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Voltage Sources in Parallel When two equal sources are connected in parallel Each source supplies half the required current

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VOLTAGE SOURCES IN PARALLEL FIG. 6.46 Demonstrating the effect of placing two ideal supplies of the same voltage in parallel.

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VOLTAGE SOURCES IN PARALLEL Because the voltage is the same across parallel elements, voltage sources can be placed in parallel only if they have the same voltage. The primary reason for placing two or more batteries or supplies in parallel is to increase the current rating above that of a single supply.

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Voltage Sources in Parallel Voltage sources with different potentials should never be connected in parallel Large currents can occur and cause damage

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VOLTAGE SOURCES IN PARALLEL If for some reason two batteries of different voltages are placed in parallel, both will become ineffective or damaged because the battery with the larger voltage will rapidly discharge through the battery with the smaller terminal voltage. FIG. 6.47 Examining the impact of placing two lead-acid batteries of different terminal voltages in parallel.

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Backup Slides

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Two Resistors in Parallel For only two resistors connected in parallel, the equivalent resistance may be found by the product of the two values divided by the sum Often referred to as “product over the sum” formula

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Current Divider Rule For only two resistors in parallel:

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Equal Resistors in Parallel Total resistance of equal resistors in parallel is equal to the resistor value divided by the number of resistors R T = R/n

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Resistors in Parallel Total resistance of resistors in parallel will always be less than resistance of smallest resistor

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